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Some remarks on multiplicity codes
Multiplicity codes are algebraic error-correcting codes generalizing
classical polynomial evaluation codes, and are based on evaluating polynomials
and their derivatives. This small augmentation confers upon them better local
decoding, list-decoding and local list-decoding algorithms than their classical
counterparts. We survey what is known about these codes, present some
variations and improvements, and finally list some interesting open problems.Comment: 21 pages in Discrete Geometry and Algebraic Combinatorics, AMS
Contemporary Mathematics Series, 201
Algebraic List-decoding of Subspace Codes
Subspace codes were introduced in order to correct errors and erasures for
randomized network coding, in the case where network topology is unknown (the
noncoherent case). Subspace codes are indeed collections of subspaces of a
certain vector space over a finite field. The Koetter-Kschischang construction
of subspace codes are similar to Reed-Solomon codes in that codewords are
obtained by evaluating certain (linearized) polynomials. In this paper, we
consider the problem of list-decoding the Koetter-Kschischang subspace codes.
In a sense, we are able to achieve for these codes what Sudan was able to
achieve for Reed-Solomon codes. In order to do so, we have to modify and
generalize the original Koetter-Kschischang construction in many important
respects. The end result is this: for any integer , our list- decoder
guarantees successful recovery of the message subspace provided that the
normalized dimension of the error is at most where
is the normalized packet rate. Just as in the case of Sudan's list-decoding
algorithm, this exceeds the previously best known error-correction radius
, demonstrated by Koetter and Kschischang, for low rates
Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes
In this paper, a lemma in algebraic coding theory is established, which is
frequently appeared in the encoding and decoding for algebraic codes such as
Reed-Solomon codes and algebraic geometry codes. This lemma states that two
vector spaces, one corresponds to information symbols and the other is indexed
by the support of Grobner basis, are canonically isomorphic, and moreover, the
isomorphism is given by the extension through linear feedback shift registers
from Grobner basis and discrete Fourier transforms. Next, the lemma is applied
to fast unified system of encoding and decoding erasures and errors in a
certain class of affine variety codes.Comment: 6 pages, 2 columns, presented at The 34th Symposium on Information
Theory and Its Applications (SITA2011
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